108156.pdf

120 Mathematics for Finance

Sincewmust satisfy (5.14), it follows thatγ∼= 3 .694 and the weights in the
market portfolio are

w∼=

[

0 .079 0.363 0. 558

]

Exercise 5.16

Suppose that the risk-free return isrF= 5%. Compute the weights in the

market portfolio constructed from the three securities in Exercise 5.11.

Also compute the expected return and standard deviation of the market

portfolio.

5.4.2 Beta Factor........................................

It is important to understand how the returnKV on a given portfolio or a
single security will react to trends affecting the whole market. To this end we
can plot the values ofKVfor each market scenario against those of the return
KMon the market portfolio and compute theline of best fit, also known as the
regression lineor thecharacteristic line. In Figure 5.12 the values ofKMare
marked along thexaxis and the values ofKValong theyaxis. The equation
of the line of best fit will be

y=βVx+αV.

Figure 5.12 Line of best fit

For any givenβandαthe values of the random variableα+βKMcan be
regarded as predictions for the return on the given portfolio. The difference
ε=KV−(α+βKM) between the actual returnKV and the predicted return